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Register a new userCircuit-Depth Reduction of Unitary-Coupled-Cluster Ansatz by Energy Sorting
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Quantum computation represents a revolutionary means for solving problems in quantum chemistry. However, due to the limited coherence time and relatively low gate fidelity in the current noisy intermediate-scale quantum (NISQ) devices, realization of quantum algorithms for large chemical systems remains a major challenge. In this work, we demonstrate how the circuit depth of the unitary coupled cluster ansatz in the algorithm of variational quantum eigensolver can be significantly reduced by an energy-sorting strategy. Specifically, subsets of excitation operators are pre-screened from the unitary coupled-cluster singles and doubles (UCCSD) operator pool according to its contribution to the total energy. The procedure is then iteratively repeated until the convergence of the final energy to within the chemical accuracy. For demonstration, this method has been sucessfully applied to systems of molecules and periodic hydrogen chain. Particularly, a reduction up to 14 times in the number of operators is observed while retaining the accuracy of the origin UCCSD operator pools. This method can be widely extended to other variational ansatz other than UCC.
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The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
Neural-Network Quantum State (NQS) has attracted significant interests as a powerful wave-function ansatz to model quantum phenomena. In particular, a variant of NQS based on the restricted Boltzmann machine (RBM) has been adapted to model the ground state of spin lattices and the electronic structures of small molecules in quantum devices. Despite these progresses, significant challenges remain with the RBM-NQS based quantum simulations. In this work, we present a state-preparation protocol to generate a specific set of complex-valued RBM-NQS, that we name the unitary-coupled RBM-NQS, in quantum circuits. This is a crucial advancement as all prior works deal exclusively with real-valued RBM-NQS for quantum algorithms. With this novel scheme, we achieve (1) modeling complex-valued wave functions, (2) using as few as one ancilla qubit to simulate $M$ hidden spins in an RBM architecture, and (3) avoiding post-selections to improve scalability.
The Quantum State Preparation problem aims to prepare an n-qubit quantum state $|psi_vrangle=sum_{k=0}^{2^n-1}v_k|krangle$ from initial state $|0rangle^{otimes n}$, for a given vector $v=(v_0,ldots,v_{2^n-1})inmathbb{C}^{2^n}$ with $|v|_2=1$. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth complexity remains open in the general case with ancillary qubits. In this paper, we study efficient constructions of quantum circuits for preparing a quantum state: Given $m=O(2^n/n^2)$ ancillary qubits, we construct a circuit to prepare $|psi_vrangle$ with depth $Theta(2^n/(m+n))$, which is optimal in this regime. In particular, when $m=Theta(2^n/n^2)$, the circuit depth is $Theta(n^2)$, which is an exponential improvement of the previous bound of $O(2^n)$. For $m=omega(2^n/n^2)$, we prove a lower bound of $Omega(n)$, an exponential improvement over the previous lower bound of $Omega(log n)$, leaving a polynomial gap between $Omega(n)$ and $O(n^2)$ for the depth complexity. These results also imply a tight bound of $Theta(4^n/(m+n))$ for depth of circuits implementing a general n-qubit unitary using $m=O(2^n/n)$ ancillary qubits. This closes a gap for circuits without ancillary qubits; for circuits with sufficiently many ancillary qubits, this gives a quadratic saving from $O(4^n)$ to $tildeTheta(2^n)$.Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide range of parameter regime. The results can be viewed as (optimal) time-space tradeoff bounds, which is not only theoretically interesting, but also practically relevant in the current trend that the number of qubits starts to take off, by showing a way to use a large number of qubits to compensate the short qubit lifetime.
In classical computational chemistry, the coupled-cluster ansatz is one of the most commonly used $ab~initio$ methods, which is critically limited by its non-unitary nature. The unitary modification as an ideal solution to the problem is, however, extremely inefficient in classical conventional computation. Here, we provide the first experimental evidence that indeed the unitary version of the coupled cluster ansatz can be reliably performed in physical quantum system, a trapped ion system. We perform a simulation on the electronic structure of a molecular ion (HeH$^+$), where the ground-state energy surface curve is probed, energies of excited-states are studied and the bond-dissociation is simulated non-perturbatively. Our simulation takes advantages from quantum computation to overcome the intrinsic limitations in classical computation and our experimental results indicate that the method is promising for preparing molecular ground-states for quantum simulation.
The Variational Quantum Eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers. However, the circuit depth is expected to grow significantly with problem size. Increased depth can both degrade the accuracy of the results and reduce trainability. In this work, we propose a novel approach to reduce ansatz circuit depth. Our approach, called PermVQE, adds an additional optimization loop to VQE that permutes qubits in order to solve for the qubit Hamiltonian that minimizes long-range correlations in the ground state. The choice of permutations is based on mutual information, which is a measure of interaction between electrons in spin-orbitals. Encoding strongly interacting spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state. For representative molecular systems, LiH, H$_2$, (H$_2$)$_2$, H$_4$, and H$_3^+$, we demonstrate for linear qubit connectivity that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy. This approach can be extended to any qubit connectivity and can significantly reduce the depth required to reach a desired accuracy in VQE. Moreover, our approach can be applied to other variational quantum algorithms beyond VQE.
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